Large time, small coupling behaviour of a quantum particle in a random field

A NNALES DE L ’I. H. P., SECTION A

G. F. D ELL ’A NTONIO

Large time, small coupling behaviour of a quantum particle in a random field

Annales de l’I. H. P., section A, tome 39, n

^o

4 (1983), p. 339-384

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339

Large time, ^small coupling ^behaviour

of

^a

quantum particle ⁱⁿ

^a

random field

G. F. DELL’ANTONIO

Istituto di Matematica G. Castelnuovo, Universita di Roma Inst. Henri

Vol. XXXIX, ^n° 4, 1983,

Section A :

Physique ^’ theorique. ^’

SUMMARY. - For a quantum mechanical

particle

^{in a} suitable random field we prove that all finite-dimensional distributions of extensive obser- vables converge in the Van Hove

limit

to the

corresponding

distributions of a classical Poisson process. In the classical limit this process converges to a diffusion.

RESUME. 2014 On

demontre,

pour ^une

particule quantique

^{dans un}

potentiel

aleatoire

convenable,

que ^toutes les distributions de dimension finie d’obser- vables extensives convergent dans la limite de Van Hove vers les distri- butions

correspondantes

^d’un processus de Poisson

classique.

^{Dans la}

limite

classique,

^ce processus converge ^{vers un} processus de diffusion.

1. INTRODUCTION

The motion of a classical or quantum system ^{in a} random environment is

expected

^to converge ^{to a} Markov _process under suitable

scaling ^limits, usually involving large

time scales and small

coupling.

^{Results in this}

direction appear in the literature under various

headings,

^e. g.

homogenei- zation, method

of _average, diffusion

limit,

^and

require

ⁱⁿ

general

^detailed

specifications

^{as to what} constitutes a random environment and which

are the observable

quantities

^to be studied.

_ Formal results and

applications

^{can be}

found,

^e. g., in

[1 ].

^The

subject

has also been considered in the mathematical

literature ; general

^results

Annales de l’Institut Henri Poincaré-Section A-Vol. XXXIX, 0020-2339/ 1983/339/$ 5,00/

(0 Gauthier-Villars 13

340 G. F. DELL’ANTONIO

can be found in

[2 ],

^{where one considers the case} in which the interactions with the random environment are a

priori

^{assumed to be}

weakly

^correlated

in time. These results can be viewed as a version of the central limit theorem for a class of

dynamical

systems.

A more

physical setting

^{is one in which} the random force field is assumed to have weak correlations in _space, to

reproduce

^the

properties

^{of a}

rapidly fluctuating

environment. In this _case, the fact that successive interactions

are

weakly

correlated in time becomes part ^{of the}

proof (and

^indeed

usually

the most difficult

part);

^{once this is}

established,

the results follow modulo

(often

very

substantial)

technical details. This more

physical setting

^is

beyond

the reach of the

general

results in

[2 ],

^and

proofs require

^the

develop-

ment of

specific techniques.

For

example

ⁱⁿ

[3] it

^is

proved,

under suitable but rather weak _assump-

tions,

^{that the}

velocity

process of a classical

particle moving

^{in a random}

force field converges

weakly

^{to a diffusion} process in the limit in which the force field becomes

(locally)

infinitesimal and the time scale is chosen

indefinitely large (Van

^Hove

limit).

Here we prove ^a similar result for the motion of a quantum

particle

ⁱⁿ

a random

potential

^field.

As in the classical _case, convergence will hold

only

^{for a} restricted class of

observables,

ⁱⁿ

particular

for bounded continuous functions of momentum.

Results in this direction are contained in a

germinal

paper

by

L. Van Hove

[4 ]. Important

steps ^and

proofs

^{are in}

[5 ], [6 ].

^The

limiting

process is here a Poisson _process, with transition

amplitudes depending

on Planck’s constant ~.

It is ^a rather obvious

question

^to

inquire

^{whether the} results of

[3] ]

for the classical case can be recovered in the classical limit. This is indeed the _{case ;} in the last section of this _paper we shall

briefly

^indicate the way in which a

proof

^is

given.

^{A full}

proof

will be contained in

[10 ].

It should be noted that the results we present ^{here for the} quantum ^case

are obtained under conditions on the force field which are stronger ^than those of

[3 ].

The results themselves are moreover weaker than their classical counterpart, ^{in so far as we}

only

prove convergence of all finite-dimensional distributions rather than convergence of processes. The

assumptions

^are stronger both because the force field is taken to be

potential -

^{this seems}

unavoidable in

Quantum

Mechanics - and to admit moments of all

orders,

^with suitable bounds in terms of the moment of order two. Some restrictions on the bounds can be lifted

by

^{more accurate}

estimates,

^but

our method of

proof

^{does not}

exploit enough

the details of the quantum mechanical evolution on the

space-time

^scale characteristic of the

problem.

In the classical _{case, many} estimates

depend

^{on a rather detailed}

description

of « most »

trajectories;

^{that the Poisson} process of the quantum ^case

Annales de Henri Poincaré-Section A

341

A QUANTUM ^{PARTICLE IN A RANDOM FIELD}

converges ^{as a} process ^to the diffusion of the classical case suggests ^that also in the

quantum-mechanical setting

^it should be

possible

^{to have}

a better control of the ^« motion of the wave

packet »

^{for most}

configurations

of the force field.

It seems however that the main drawback of the method

presented here,

^{both in terms of}

assumption

needed and of results which can be

obtained,

^{is to be found in the fact that we are able to use}

only

^a very modest amount of

probabilistic techniques.

^In

particular

^{we lack the}

inequalities

for conditional

expectations

^{and the}

resulting tightness

^{of a suitable}

family

of

probability

^{measures, which are the main tool in the}

analysis given

ⁱⁿ

[3 ],

for the classical case.

A better strategy ^of

proof

^{could come from a more}

probabilistic approach

to the

quantum-mechanical

^case, for instance a formulation of the motion of a quantum

particle

^{in a}

potential

^{field in terms of}

integrals

^{over suitable}

functionals of a Poisson process, ^as

developed

ⁱⁿ

[8 ].

^{In this} case, it is conceivable that ^{a «} small » set of

trajectories

^will

give

^{the dominant}

contribution in the Van Hove

limit,

^{and that the}

techniques developed by

Donsker and Varadhan

[9] ]

^could put ^{to use here.}

We are indebted to Ph. Combe for some very

suggestive

discussions

on this

possibility.

The content of this _paper is as follows.

In this section 2 we

give

^{some further}

qualitative

^{comments and the}

description

^{of the} quantum mechanical evolution of a suitable class of observables in a

properly

defined random

potential

^field.

In section 3 we

provide

motivations and describe the limit Markov process.

In section 4 we

begin

^the

proof

^of convergence of the

averaged dynamics

when the

potential

^{is a Gaussian random} field and outline the strategy;

further technical details and the

completion

^{of the}

proof

^are

given

ⁱⁿ

section 5.

In section 6 we outline the

proof

^of convergence for all finite-dimensional distributions. We also outline how the

proofs

^can be extended to cover

the case of random

potential

fields which are not

gaussian.

^{In section 7}

we prove that the Markov process described in section

3,

converges, when ~ -~

0,

^{to the diffusion} process of the classical case.

2.

QUANTUM

^EVOLUTION

IN A RANDOM HOMOGENEOUS POTENTIAL FIELD Let

V(x)

^{be the}

potential

^{field. The motion of a} quantum

particle

^is

described

by

^the

Schrodinger equation

Vol. XXXIX, ^n° 4-1983.

342 G. F. DELL’ANTONIO

when A ⁼

/ ~2 ~x2i, 03C8

^E

^{L 2(1R3)}

ⁿ

^{D( - h20394}

⁺

^V),

and for convenience we 1

have taken units of mass such that m ⁼⁼

1/2.

We shall consider

only

^{motion in}

1R3, although

^all the results we state also hold &#x3E;

3,

^{and in fact some of the}

proofs

ⁱⁿ

§ 4,

^{5 become}

simpler.

Some crucial estimates in

§ 4, 5

fail instead for n ⁼⁼ 1 or

2,

^{as will be} apparent in the

sequel.

The result could still be true for n ⁼

2,

^{which is somewhat}

a borderline _case, but the method of

proof

^we present here fails in this

case. Since h

plays

^no role until

§ 7,

^{we shall set h = 1} until then.

In order that

(2.1) provide

^a

unitary

^{evolution in}

L 2([R3)

it is sufficient

by

^Stone’s

theorem,

^{that - A} + V be

self-adjoint.

If this is the _case, let be the

corresponding

one-parameter group of

unitary

operators ;

one has ⁼ _exp

i( -

^{1B +}

V)t.

Let B be a

symmetric

bounded linear operator ^on

LZ(f~3),

^{i. e. a} quantum mechanical

observable ;

^{its time}

evolution,

^{in the}

Heisemberg representation (which

^{we shall}

adopt)

^is

given by

Let

R3

_{E a} ^-H-

V(a), (V(~)~)(~) = .p~ - Q)

^{be the standard}

representation

of the _group of

space-translations.

We shall denote

by ~o

^{the linear} span ^over the

complex

^{numbers of}

the observables which commute with for

all a

^E

^{R 3.} j~o

^is

easily

^seen

to be a commutative C*

algebra,

^{which can} be identified via Fourier transform with the

algebra

^of

essentially

bounded functions on [RP.

Indeed,

^{if A E one has}

for some function

A(p_)

^E

L 00([R3). Here ~

^{is the} Fourier transform of

1/1.

We shall call this the Fourier

representation

^of

Ao.

^Denote

by Co(R 3)

the class of continuous functions which vanish at 00; is a

subalgebra

of

L 00,

^closed in the supremum ^norm. Let s~ be the

subalgebra

^of

j~o

which has as

representative

^{in the Fourier}

representation;

^{~ is}

then closed in the norm

topology.

^The observables for which we shall prove limit theorems ^are the

symmetric

elements of j~.

We recall now

briefly

^the definition of ^a random field. Let Q be a

proba- bility

space, with

generic point

^~ endowed with the measure ,u. Let S be

a linear

subspace

^of

C(R 3; R) (continuous

^{functions from}

^R3

^to

R)

^and

let S 3

f ~

^{be a} linear map from S to the linear _space of random variable

(~-measurable

^{functions over}

Q).

Formally,

^{one writes} ,, t/

where for

co)

^{is a}

(generalized)

^{function of jc. In favourable}

Annales de l’Institut Henri Poincaré-Section A

343

A QUANTUM PARTICLE IN A RANDOM FIELD

cases,

úJ)

will be for each x a random variable. For a measurable and

integrable

^{function on}

Q,

^{we define ,}

We

require

^{that V be}

stationary

^and

ergodic.

The field

V(x)

^is

stationary

^if

for

all a

^E

[R3,

^{as elements of}

(S*)n,

^S*

being

^{the dual of S.}

One can choose Q in such a way that there exists a

representation

^of

^R3 by unitary

operators

Ta

^on

L 2(0, ,u)

^{such that}

Ergodicity implies

that every measurable function of the random field

V,

which is invariant under

Ta

differs from a constant function

only

^on

a set of _~-measure zero.

If E(V(xl)

^{... exist as} continuous

functions,

^then for each x~

1R3,

is a random

variable,

^{and one can choose a} modification

(on

^{a set}

of zero

measure)

^of such that the

resulting

^field

is jointly

^measurable

in x and úJ.

These conditions are in

particular

^met

if V(x)

^{is a centered}

( =

^mean

zero)

Gaussian field of covariance

~(ç),

where G is continuous. One has then of course

We shall state our results and

give proofs only

^{in the case in which}

V(x)

is a Gaussian random field. As will become apparent ^{in the course of the}

proofs,

^{the results can} be extended to more

general

^random

fields, provided

one has suitable a

priori

^{bounds on the moments of V.}

On the Gaussian random field V we shall make the

assumption.

ASSUMPTION A. ^-

If |g|1

^{is the L 1 norm}

of g,

^{we shall use the notation}

Having

^{thus set our}

notation,

^we

begin constructing

^the evolution of the observables in j~ under the influence of the random

potential

^{field V.}

In the Gaussian _case, it is not difficult to prove that there exists a set

Qo

of measure one, such

that,

^{if úJ E}

03A90, H( úJ) == - 1B

⁺

co)

^is

essentially self-adjoint

^on

CÜ(1R3).

^{This is} sufficient to define a

dynamics

^for a . a . cc~.

We shall however be interested in

regularity properties

^of the average

Vol. XXXIX, n° 4-1983.

344 _{G. F.} DELL’ANTONIO

dynamics.

^{To obtain}

these,

^{we choose to}

approximate

^{first and}

to define the

dynamics by

^a

limiting procedure.

^A natural choice would be For all _{x, ~} is bounded

below, uniformly

ⁱⁿ x, ^co. If it follows from the individual

ergodic

^{theorem that}

for a . a . cc~,

VE1~( ~, cc~) E L o~(~3),

the space of functions which are in L2 when restricted to any bounded subset of

[R3.

Therefore there exists a set

03A91

^c

Q, (03A91)

⁼

1,

^such

that,

^if

- Ll +

VE1~( ~,

^is

essentially self-adjoint

^on

CÜ(1R3).

^Moreover

is a core for - 0394 +

V(1)~, and exp i(- 0394

⁺

V(1)~(., 03C9))t is strongly

^continuous

in t for all cc~ E

Q,

^and

strongly

measurable in cc~ for all t

(this

^{can be}

proved,

e. g.,

using

the Trotter

product formula,

^{since the}

pointwise

^{limit of measu-}

rable functions is itself

measurable).

While

(2. 5)

^is in many ways ^a natural

approximation,

^it

requires

^much

machinery

^to prove

that,

^{for all t E}

R,

^{úJ E}

03A91,

the limit ~~0 exists as a

unitary

operator.

We shall choose therefore the

following approximate

random field Since is

jointly

measurable in

(x, a~),

^{so is Moreover}

by construction, V(x, úJ)

is bounded

uniformly

ⁱⁿ

(x,

We will prove

LEMMA. 2014 For

each t, 03BB

^{there is a}

set Q

^c

^{Q, (03A9)’}

⁼⁼

^1,

^{and a sequence ~n,}

8n ^~ 0 when ~ -~ _oo, such

that,

^{if úJ E}

Q,

strong

limit

⁺

exists. Call this limit.

_

Then

U~(~)(~)

^is

unitary

^{for all úJ E Q and}

~-measurable.

D

Proof

^{2014 We} shall prove

that,

^for

every ~

^{one has}

Assuming

^{for the moment the}

validity

^of

(2.7),

^we

complete

^the

proof

of the Lemma.

be a denumerable basis in

L 2([R3).

^From

(2.7),

is,

^{for each K =}

1, 2,

^{... a}

Cauchy

sequence in

L 2([R3

^X

^Q, v),

^where

v = ,~~ x ,u and is

Lebesgue’s

measure on [R3. It then follows

that,

^for

each K E Z + there is a set

= 1,

^{and a}

subsequence 0,

^such

that,

^{for all cc~ E A +}

~K

converges in

Let Q ⁼ Therefore ⁼ 1.

KEZ+

Annales de l’Institut Henri Poincaré-Section A

345

A QUANTUM ^{PARTICLE IN A RANDOM FIELD}

One can choose a

sequence ~n ~

^{0 such}

^that,

^{if 03C9 E}

^Q,

converges in for all

(the sequence {~}

^{is a}

subsequence

of

each { ).

Since + is norm-bounded

uniformly

^{in 8,}

converges in for

all 03C8

^{E as can be seen}

approximating 03C8

^with

finite linear combinations of the Let be the limit

point.

^From

(2 . 7)

it follows

that 03C8 ~ ~03C8(03C9)

is linear and

bounded,

^{and in}

fact ~~03C8(03C9)~ = II 03C8 II I (all

^norms

being L 2([R3) norms)

^{since the unit}

sphere

is closed under _sequen-

tial strong convergence. _ _ _

Therefore for

each t, aL

there exists a

set

^{Q c}

^{Q, ~(Q)}

^{= 1}

^(the

^{set Q}

depends

ⁱⁿ

general

on t,

03BB)

^such

that, if 03C9~03A9,

^{there exists a}

unitary

operator which is the strong ^{limit of + t.}

Measurability

of follows since it is the

pointwise

limit of measurable functions.

It remains therefore to prove

(2.7),

^{which in turn is}

equivalent

^to

We shall use the

following identity,

where the series is norm convergent for all ~ &#x3E; 0 and úJ E

Q, uniformly

in co.

We shall refer to

(2.8)

^{as «}

Dyson

^{series ».} Notice that the left-hand of

(2. 8)

^{satisfies the} differential

equation

where

V,(t)

⁼

The series

(2. 8)

^{is obtained}

by iterating

^the

integrated

version of

(2.9),

also called Duhamel’s

formula,

^{or «} variations of constants ».

Substituting (2. 8)

ⁱⁿ

(2. 7’)

^{one sees that one must}

study

^{the limit when}

~

^{~ 0 of}

Vol. XXXIX, ^{n° 4-1983.}

346 G. F. DELL’ANTONIO

(the exchange

^{of the summation} over m, n and

integration

over ti, Tj is

legitimate

^{in view of the} boundedness of the The

integrand

^{can be}

expressed

^{as a} formal series

Let _Eo ⁼ max

(8, 8’). Using

^the

properties

^{of the Gaussian}

integrals,

we shall prove that

380

^such

that,

^for Eo

Eo,

^{the series}

(2.11)

^is

absolutely

convergent,

uniformly

ⁱⁿ 80. From this result

(2. 7’) follows,

^{since it is}

easily

verified that

(2.10)

^{has no terms of order zero in} Eo.

In

particular,

^{one has}

and the sum is over all unordered

pairing

^{of the}

points {xi},

^{each of which}

is taken with

multiplicity 2KI

^{+ 1. In}

(2.12), Kmn E Z+

^{is the number of}

the times the

point

xm is

paired

^{with the}

point

xn.

By carrying

^out

explicitly

^all calculations one verifies that each

integrand

in the serie

(2.10) gives

^a contribution which is bounded in absolute value

... Y ..

by ~ A~2~j~1

¹

j= 1 2 11.p 112 independently of t

1 ^... zm. The inte-

gration

over t 1 ^... 2m

provides

^{for each such term a factor}

tn + m(n !) -1 (m !) -1.

To prove absolute convergence of

(2.11), uniformly

^{in 0 ~ is}

therefore sufficient to prove absolute convergence of the series

where represents ^{the number of}

pairing

among n ^{+ m}

points

xi, i ⁼⁼ 1 ... n + m, each taken with

multiplicity Ki,

^{and we have set}

= ~

~=1...~.

To evaluate

N,

it is easier to count

pairings

^{in a somewhat different} way.

Annales de l’Institut Henri Poincaré-Section A

347

A QUANTUM ^{PARTICLE IN A RANDOM FIELD}

Let

Kij ^{i, j}

^{= 1 ... n + m} be the number of times the

pair (i, j)

appears in the

pairing. Obviously

The number of

pairing

^{is then}

We rewrite then

(2.13)

^as

We now use the fact

that,

^if

(2.13) holds,

^then

and repeatedly

^Schwartz’

inequality

^{to dominate the series in}

(2.16) by

where 2K, ^{+ 1 ==} 2K.. + K,;.

Now,

and

moreover

Therefore

(2.17)

is dominated

by

Vol. XXXIX, ^{n° 4-1983.}

348 G. F. DELL’ANTONIO

which is

absolutely

convergent

uniformly

ⁱⁿ

^/),,

^{~ over} bounded sets for

Bo2-5!2(11~lll)-1.

^D

We are

mostly

interested in the random evolution of quantum observables.

This is

again

^defined

by

^a

limiting procedure.

Let

We have

THEOREM 1. 2014 For

each t,

^{~, there is a set} Q ^c

Q, ~u~) - 1,

^{and a} sequence 0 such

that,

if 03C9~03A9 and strong

limit A~n,03BB(t, ^úJ)

exists. Call the limit. Then A -~ is for each úJ a

unitary isomorphism

of A with a

subalgebra of B(L 2([R3)).

^{Moreover is}

weakly

,u-measu- rable for

each t,

^/).

and,

^for

Proof -:-

^A part ^from

measurability

^and

(2 . 20),

^all

remaining

^statements

of Theorem 1 follow from Lemma

1,

^and moreover one has

To prove the

remaining

^two statements, ^{it suffices to} prove

that,

^for

any

given A E ~, ~ E L2(~3), ~,,

^{t E R.}

Indeed,

^{from the} _strong convergence of to

A~{t,

^for

and Schwartz’s

inequality,

^{one concludes that for}

every ~

^E

L 2(1R3) (03C8, A~n,03BB(t)03C8)

converges in

,u)

^to

(1/1, A03BB(t)03C8)

^{and this}

implies

^measurabi-

lity and (2.20).

The

proof

^of

(2.21)

^{follows the same lines as the}

proof

^of

(2.7).

^One

starts from the

Dyson series,

^obtained

by iterating

^the

integral

^version

of the

equation

where

A£,~(t) -

The

Dyson

series for observables is

a norm convergent ^{series in view of the} definition

ofV,.

^The

proof

^of

(2.21) given (2 . 22)

^{follows then the same} steps ^{as the}

proof

^of

(2 . 7)

^{in Lemma 1,}

and we shall not repeat ^{the details here.} D

Annales de Henri Poincaré-Section A

349

A QUANTUM ^{PARTICLE IN A RANDOM FIELD}

Remark. 2014 Notice that Theorem 1 and its

proof provide

^{also an}

explicit

formula for

E(.p,

^{One has indeed}

From the

proof

^of theorem 1 it also follows

COROLLARY. For every choice

of t 1,

^..., tn, ^{~, there} is a set

Q’,

⁼⁼

1,

and a

sequence ~n ~ ^0,

^{such that if co E Q’ and E A i = 1 ... n, the}

strong ^{limit of}

AEm},~(tl) ~ ... ~ AEm~,~(tn)

exists and coincides with

Moreover the latter operator ^is

weakly

measurable in co, and

for D

- Since the

are uniformly bounded for

^{£ 0 and}

weakly measurable,

^the operators ^{= and}

A~)

⁼

E(A~(~))

are well defined and

belong

^to

j~o

^since the process is

stationary.

Similarly, ... ~ A~,n~(tn)) E ~o,

but it is of course different from

A~,l ~(t 1 )

^...

We now prove that all these operators ^{are in fact in j~. Indeed one has}

LEMMA 2. - For all choices ^...

and

is jointly

continuous in the

tk’s.

^In

particular

the average

dynamics

A ~

A~(t)

is defined in j~ and continuous in t. D

Proo, f.

^shall

give

^the

proof only

^{for n = 1. A} part from notational

complications,

^{there is no}

difficulty

ⁱⁿ

extending

^the

proof

^{to the}

general

case.

From

(2.22)

^{one obtains in the Fourier}

representation

Vol. XXXIX, ^n° 4-1983.

350 G. F. DELL’ANTONIO

where _po ⁼⁼ _{pn = ~} and the third sum is over all

permutations K iK

such that

tiK+

^{1 if}

K hand tiK tiK+

^{1 if K} &#x3E; h.

The estimates

given

^{in the}

proof

^{of Lemma 1 can then be}

applied

^to

(2 . 25) (there

^{is an extra factor 2" in the}

estimates, coming

^{from ;}

^indeed

^this

perm

sum

corresponds

^to

summing

^{over the 2n terms in the}

multiple

^commutator

which

appears

ⁱⁿ

(2. 22))

^and

they

^{are uniform in}

Therefore _converges to

A03BB(t)

^{in norm}

(recall

^{that for}

each 03C9~

one had

only

strong

convergence)

and it remains to prove that ^E j~.

By

^{the same estimates as}

above,

^{the series}

(2. 25)

^is

absolutely

convergent

uniformly in p,

and therefore it suffices to prove that .each term

belongs

to ~.

Each term in

(2.25)

^{with h ~}

0, n

^{can be} put in the form

where K n C

(and depends parametrically on ~,, t),

^{while the terms}

with h ⁼ 0 or n

(and

^therefore ph ⁼

p)

^are of the form where

K 1

^{is a bounded} continuous function

of p.

Since A E

~, clearly K 1

^{A E j~.}

To prove that ^E ~~, notice first that

B(/?)

^is continuous, since K E L 1.

Indeed, for all

y, ; y ;

&#x3E;

0

^{and N} &#x3E; 0 one has

,

and this

expression

^{can be made}

arbitrarily

^small

by

^first

choosing

^N

sufficiently large

^and

then I ~ I sufficiently ^small, using

^the

continuity

To prove

that lim ^B(p)

⁼

^0,

^notice

^that,

^{since A E}

^{~, given ~}

&#x3E; 0

-

~

there exists

M£

^such

that if p ~

&#x3E;

M£, where K 11

^{is the} L2-norm of K.

-

1

-

One has

then, for 1£

&#x3E;

Nt

⁺

Mt

since,

if

^I

Ne and 1£

&#x3E;

Ne

⁺

then 1£ - £’

&#x3E;

Me.

Q

Annales de l’Institut Henri Poincaré-Section A

A QUANTUM ^{PARTICLE IN A} RANDOM FIELD 351

It follows from Lemma 2 that A ~ is a linear continuous map of A into itself. It is

given explicitely

ⁱⁿ

(2. 23)

^{as a} norm-convergent power serier in

/L,

^{but it is in}

general

^not differentiable

in t;

^{even for those A E j~}

for

which differentiability

^{can be}

proved,

^no

simple equation

will be satisfied

by A;.(t).

^. _

We shall however prove in

94

^that

A;.Cr/Å2)

converges, when ~, ~

0,

to a Markov

semi-group

with continuous parameter ^r.

3. SOME PROPERTIES OF THE EVOLUTION IN

‘

THE LIMIT

t = ~/~,2,

^/~0

We shall

study

ⁱⁿ

§

^{4 the limit t =}

z/~,2, ~, ~

^{0 of the}

averaged dynamics

and of all correlation functions.

Here we

provide

^some motivation to indicate which is the limit to be

expected,

^{and we}

study

the convergence of a sequence of Markov processes

T).

somewhat related to the

averaged dynamics.

A part ^from

giving

^{some hints at the} mechanism which will be put

at work in

§ 4,

^we

give

^{here also some} estimates which will be of use in the

sequel.

As a

preliminary

^{we shall}

study

^the operator

Obviously

Formally, A~~~/~,2~

^{satisfies the}

equation

where

Ho - - 4.

This relation is

only suggestive,

^{since we have not}

proved

that there

are cu E Q for which

A~(~

^is

differentiable,

^{even if}

only weakly.

We write

formally

and substitute in

(3.1), equating

^terms of the same order in ~,. This leads to

The first relation in

(3 . 2)

^is

compatible

^with

A(0)(t, úJ)

^E

A0

for all 03C9 ^~

SZ1, although

^{it does not}

imply

^{it. Due to the}

ergodicity

of the process, ^one has then

A~°~(t,

⁼

E(A~°~(t))

^for

Vol. XXXIX, ^{n° 4-1983.}

352 G. F. DELL’ANTONIO

From the second relation in

(3.2)

^{we conclude}

then,

^{at a} formal level

(since

^{= 0}

1).

--

Equation (3.3)

is understood in the sense

that,

^for

all 03C8

^{for which the}

integral

converges for _{a. a .p,}

where 03C8

^{is the} Fourier transform

of 03C8

^E

L 2(1R3)

^and

V(/7, p’ ; 03C9)

is defined

We now substitute

(3.3)

^{in the} third relation in

(3.2), again formally

since we do not control the domains of the generators

involved,

^{to obtain}

We

integrate

^over

Q,

^{and use the fact}

that,

^{due to the}

stationarity

^{of the}

process,

E( [Ho, A(2)(T)]) == 0,

^{and moreover is}

independent

of co on a set of measure one. One obtains

where

Therefore,

Remark. ^- This heuristic argument ^would suggest ^{that a} stronger result should be

expected, namely

that there is a set

03A91

^c

Q,

⁼

1,

such that

for all 03C9~03A91.

We do not know if

(3.7’)

^{holds in a weak} sense, with

Q1 depending

on T and ^on the vectors in which enter in the definition of weak convergence.

Certainly (3 . 7’)

^{cannot hold in a} strong sense, since it would contradict the result we establish for

Annales de Henri Poincaré-Section A

353

A QUANTUM PARTICLE IN A RANDOM FIELD

The derivation of

(3 . 7) given

^{above is at best}

heuristic,

^as evident from the remark above.

Still, (3 . 7)

^is correct, ^{as we shall} prove in

94.

We shall do so

by approximating E(A;.(T/Â2) by

^{the solution}

A~(r)

^of

a suitable linear

equation.

In

g4

^we shall prove that

E(A~(T/~)) 2014

converges ^{to zero} in the

topology

^{of j~} when )" ^~ 0. Here we shall define

A~(r)

^and prove

that, again

^{in the}

topology

^of

~,

converges ^to when ~ -~ 0.

We

begin by noting

^that

when

~ÀCr/)~2)

⁼ exp

T/À2)AÀ(T/)~2)

exp

( - i~~/~~2)

^and

^that, according

to

(2 . 23)

where

From

(3.9)

^{one should} expect ^{~ 0} when /t -~

0,

^{due to some mild}

mixing properties

^of the process and the local

decay

^{of for}

large

^t.

Indeed,

^{this is what is}

proved

^{in the classical} case,

using a priori

estimates for conditional

expectations

^{and some} information on the

properties

^{of « most » classical}

trajectories.

^We shall prove

in § 4

^that

D~(~)

^{~ 0 in the}

topology

^{of j~. Motivated}

by this,

^{we define to}

be the

(unique)

solution of

Notice that is « sure », i. e. it does not

depend

^{on co.}

Eq. (3.10)

^{can be solved}

by iteration,

^which

provides

^{a norm} convergent series for all

~,,

^{T. The} solution is therefore

unique,

^{and this} proves that

since,

^if

A{ 1 ~(z)

^{is a}

solution,

^{so for}

In the Fourier

representation

^{one has}

explicitely

Let J~f be defined as in

(3.6)

^{and let} exp - ~t be the

semi-group

^it generates

(the

^{existence of}

exp ( -

^is part ^{of the}

proof

^{of the next}

lemma).

Vol. XXXIX, n° 4-1983. ^~ ,

354 G. F. DELL’ANTONIO

One has then

LEMMA 3. 2014 For all and r ^~

0, A(T) - ^{e - ~z.}

^A converges ^to zero, when ~, ~

0,

^{in the}

topology

^{of j~.} D

Proof

^{2014 From}

(3.11)

^{it is}

easily

^seen

that

^{for some 0.}

We write

(3.11)

ⁱⁿ

integro-differential form, taking

^the derivative with respect ^to T, and then take

Laplace

^transform.

For ,u

&#x3E; Yo, define

From

(3.11)

^{one has}

with

As a linear operator ^on

~0(~3), ~~,;,~

is accretive for all _,u &#x3E; 0.

Indeed,

if

f

^E

Co([R3),

^let

_{p_o}

^{be a}

point

^at

which

reaches its maximum

(all

^our

function _spaces are

real).

Let lf

be the element of defined

by

⁼ We prove

that 0.

Indeed,

^{notice that} == 0 =&#x3E;

f

⁼

0;

^{assume then}

0. Then

Since, by

definition of

/(~o)) ~

^{0 VK E}

[R3,

^{one has}

indeed

!/~/) ~

⁰

-

Since ~ is

bounded,

range

(~

⁺

~o) _

^for

~,o sufficiently large.

We conclude

that ~~

is accretive for _{every ,u} &#x3E; 0. Therefore its spectrum is contained

in {ç ^{Re 03B6} 0}

^{and we can}

^define,

^for

^all

&#x3E;

0,

Clearly A~

⁼

A~;~

^{at least}

for ,u

&#x3E; ~ since

A~;~

^is

^analytic

^for

^0,

it follows that

A03BB(t)

^{has a}

Laplace

^transform

for

&#x3E;

0,

^and

A03BB; = A03BB,

&#x3E; 0. Let be defined

by

⁼

(2 -

^/~’

I) -1 A{ p) ;

^{since ~ is}

accretive

(the proof

^is

given

^as

above) A~~~{p)

^{is well} defined for all _,u, Re _~c &#x3E;

0,

and is in fact the

Laplace

transform of

From

(3.15),. (3.14)

The

right-hand

^side converges ^{to zero} in the

topology

^of

~ ; since,

^for

,u &#x3E;

0,

^{I -}

~~,,~)

^{is bounded away from zero}

^uniformly

ⁱⁿ ~, &#x3E;

0,

^we conclude

A03BB, - A( ) ~

^{0 for}

^all

&#x3E;

0, uniformly

^{in 0 ð} ,u N.

Annales de l’Institut Henri Poincaré-Section A

355

A QUANTUM ^{PARTICLE IN A RANDOM FIELD}

From this it follows that

A~) -

^{0 in the same}

^topology.

^D

Some information on the structure of the limit

semi-group Tt

⁼

^~~

is

provided by

^the

following

LEMMA 4. ^- The

semi-group Tt

^{is a} contraction

semi-group

^{on j~.}

It is reduced

by

^{each ball}

Ba3~ : ~ p ~ ~ p ~ I a ~

and defines on each

sphere S~3) : {£ I I p I == ~ }

^a contraction

semi-group.

^{On each}

sphere,

^{the constant}

function is left invariant

by T~.

Proo, f :

^have

already proved

^{that 2} is accretive. Therefore

Tt

^is

a contraction

semi-group. Next,

^{we notice}

that,

^if

f

^{E has} support contained

a p ~ b ~

^{for some}

^{a, b}

&#x3E;

0,

^then

Ttf

^has support contained in the same set. This is evident from the definition of

J?,

^and

the fact that is

given by

^a norm-convergent ^{series in t.}

Consider now on

(continuous

^{functions on the}

sphere

^{of radius}

one in

1R3)

^the

family

^of operators

where a &#x3E; 0 and

p) _ ap).

One verifies

easily that,

^for

all f

^E

where ^~

f a(~) _ ,

The operators

2 a are accretive for every a

&#x3E; 0. Let be the associated

semi-group.

^From

(3.18)

one verifies that

in the natural

decomposition

^of

Cü([R3B0)

^{as a}

subspace

^of

0

Constant functions on are left invariant

by

^since

~a ~

^{1 - 0.} D Assume now

that a

satisfies for _every a &#x3E; 0 a strong form of Doeblin’s

condition,

^{i. e. for}

every ~

^E

Sa3~,

^{B c}

Sa3~, ~an~( p,

^{0 for some}

B

cient condition for this to hold is that

~(~)

^has compact support, ^or exponen- tial

decay.

One has then

COROLLARY. 2014 Under the condition stated

above,

^{the constant function}

on is a

global

^{attractor for the}

semi-group Tt.

^Given any function

Vol. XXXIX, ^n° 4-1983.

356 ^G. F. DELL’ANTONIO

g ^E

Co, Ttg

converges when t ^-~ oo

towards g,

^where

g

^{is a function}

only

Proof

2014 Under the stated

conditions,

^{1 is the}

only eigenvector

^of

T~a~

to the

eigenvalue

^{zero. The}

corollary

^{follows then} from standard

properties

of contraction

semi-groups.

D

2014 The

physical description

^{behind Lemma 4} is that the random force

field,

^{in the limit in} which ~, ~

0,

^{does not alter}

appreciably

^the

energy of the

particle,

^{even if it acts for a time of order}

^{~,- 2.}

^{On such a}

long

time scale however the momentum of the

particle undergoes changes,

in such a way

that,

^{on a still}

longer

time scale

(r

^~

oo),

^its distribution becomes uniform on the mass shell.

4

CONVERGENCE

^{TO THE} MARKOV LIMIT:

CONVERGENCE OF DYNAMICS

Let be defined as in Theorem

1;

^{in this} section and in the

following

we shall

study

^{the limit of ,}

when ~, -~

0,

^{when E j~ i = 1} ... n, and T 2 ~ ^{... ~ In this} section we consider the case n ⁼

1,

and prove

where

T,

is defined as in

(3.7), (3.6).

This will be the content of Theorem 2.

Before

stating

the theorem

precisely,

^we

perform

^some

preliminary operations

^and prove ^two lemmas which will be used in the

proof

^of

Theorem 2.

From

(3.8), (3.10)

it follows

that, setting

where is defined ^as in

(3.9)

If one can prove that

D~r) -~

0 when}1. ~

0,

^then

(4.1)

^{follows from}

Lemma 3 and ^an

application

of Gronwall’s

inequality.

If one had for the present

quantum-mechanical setting

the definitions

-

Annales de l’Institut Poincaré-Section A